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Displaying similar documents to “Core Index of Perfect Matching Polytope for a 2-Connected Cubic Graph”

On Fulkerson conjecture

Jean-Luc Fouquet, Jean-Marie Vanherpe (2011)

Discussiones Mathematicae Graph Theory

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If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples....

Cores, Joins and the Fano-Flow Conjectures

Ligang Jin, Eckhard Steffen, Giuseppe Mazzuoccolo (2018)

Discussiones Mathematicae Graph Theory

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The Fan-Raspaud Conjecture states that every bridgeless cubic graph has three 1-factors with empty intersection. A weaker one than this conjecture is that every bridgeless cubic graph has two 1-factors and one join with empty intersection. Both of these two conjectures can be related to conjectures on Fano-flows. In this paper, we show that these two conjectures are equivalent to some statements on cores and weak cores of a bridgeless cubic graph. In particular, we prove that the Fan-Raspaud...

Perfect connected-dominant graphs

Igor Edmundovich Zverovich (2003)

Discussiones Mathematicae Graph Theory

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If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number γ c ( G ) of G. A graph G is called a perfect connected-dominant graph if γ ( H ) = γ c ( H ) for each connected induced subgraph H of G.We prove that a graph is a perfect connected-dominant graph if and only if it contains no induced path P₅ and induced cycle C₅.

Mácajová and Škoviera conjecture on cubic graphs

Jean-Luc Fouquet, Jean-Marie Vanherpe (2010)

Discussiones Mathematicae Graph Theory

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A conjecture of Mácajová and Skoviera asserts that every bridgeless cubic graph has two perfect matchings whose intersection does not contain any odd edge cut. We prove this conjecture for graphs with few vertices and we give a stronger result for traceable graphs.

A note on pm-compact bipartite graphs

Jinfeng Liu, Xiumei Wang (2014)

Discussiones Mathematicae Graph Theory

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A graph is called perfect matching compact (briefly, PM-compact), if its perfect matching graph is complete. Matching-covered PM-compact bipartite graphs have been characterized. In this paper, we show that any PM-compact bipartite graph G with δ (G) ≥ 2 has an ear decomposition such that each graph in the decomposition sequence is also PM-compact, which implies that G is matching-covered

Dense Arbitrarily Partitionable Graphs

Rafał Kalinowski, Monika Pilśniak, Ingo Schiermeyer, Mariusz Woźniak (2016)

Discussiones Mathematicae Graph Theory

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A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with [...] ‖G‖ > (n−42)+12 | | G | | > n - 4 2 + 12 edges is AP or belongs to few classes of exceptional graphs.

Conditions for β-perfectness

Judith Keijsper, Meike Tewes (2002)

Discussiones Mathematicae Graph Theory

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A β-perfect graph is a simple graph G such that χ(G') = β(G') for every induced subgraph G' of G, where χ(G') is the chromatic number of G', and β(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., δ(H)+1). The vertices of a β-perfect graph G can be coloured with χ(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden...

Some remarks on Jaeger's dual-hamiltonian conjecture

Bill Jackson, Carol A. Whitehead (1999)

Annales de l'institut Fourier

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François Jaeger conjectured in 1974 that every cyclically 4-connected cubic graph G is dual hamiltonian, that is to say the vertices of G can be partitioned into two subsets such that each subset induces a tree in G . We shall make several remarks on this conjecture.

On perfect and unique maximum independent sets in graphs

Lutz Volkmann (2004)

Mathematica Bohemica

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A perfect independent set I of a graph G is defined to be an independent set with the property that any vertex not in I has at least two neighbors in I . For a nonnegative integer k , a subset I of the vertex set V ( G ) of a graph G is said to be k -independent, if I is independent and every independent subset I ' of G with | I ' | ≥ | I | - ( k - 1 ) is a subset of I . A set I of vertices of G is a super k -independent set of G if I is k -independent in the graph G [ I , V ( G ) - I ] , where G [ I , V ( G ) - I ] is the bipartite graph obtained from G by deleting...